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arXiv:1110.2494 · SIAM Journal on Computing

Spectral Gap Amplification

R. D. Somma, S. Boixo

Published Jan 1, 2013• 50 citations
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Abstract

Many problems can be solved by preparing a specific eigenstate of some Hamiltonian $H$. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of $H$ for that eigenstate and the cost of evolving with $H$ for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian $H'$ with the same eigenstate as $H$ but a bigger spectral gap, requiring that constant-time evolutions with $H'$ and $H$ are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when $H$ satisfies a frustration-free property and give $H'$ for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states, which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.

DOI
10.1137/120871997

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Many problems can be solved by preparing a specific eigenstate of some Hamiltonian HHH. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of HHH for that eigenstate and the cost of evolving with HHH for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian H′H'H′ with the same eigenstate as HHH but a bigger spectral gap, requiring that constant-time evolutions with H′H'H′ and HHH are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when HHH satisfies a frustration-free property and give H′H'H′ for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states, which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.
The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of the order 1/L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upper bound is of the order 1/L2. Our proofs use recent results on adiabatic state transformations, spectral gap amplification, and the simulation of continuous-time quantum query algorithms. They also consider a reduction from the unstructured search problem, whose lower bound in the oracle cost translates into the upper bounds in the gaps. The impact of our results is that improving the gap beyond that of standard constructions (i.e. 1/L2), if possible, is challenging.
Many problems in science could be solved by preparing the low-energy quantum state (or any eigenstate) of a Hamiltonian. A common example is the Boolean satisfiability problem, where each clause can be mapped to the energy of an interacting many-body system, and the problem reduces to minimizing the energy. In quantum computing, adiabatic quantum state transformations (ASTs) provide a tool for preparing the quantum state. ASTs are conventionally implemented via slow or adiabatic perturbations to the Hamiltonian, relying on the quantum adiabatic theorem. Nevertheless, more efficient implementations of ASTs exist. In this talk I will review recently developed methods for ASTs that are more efficient and require less assumptions on the Hamiltonians than the conventional implementation [1,2]. Such methods involve measurements of the states along the evolution path and have a best-case implementation cost of L/G, where L is the length of the (evolved) state path and G is a lower bound to the spectral gap of the Hamiltonians. I will show that this cost is optimal [3] and comment on results of the gap amplification problem, where the goal is to reduce the cost by increasing G [4].
Solving for quantum ground states is important for understanding the properties of quantum many-body systems, and quantum computers are potentially well-suited for solving for quantum ground states. Recent work [1] has presented a nearly optimal scheme that prepares ground states on a quantum computer for completely generic Hamiltonians, whose query complexity scales as δ−1 , i.e. inversely with their normalized gap. Here we consider instead the ground state preparation problem restricted to a special subset of Hamiltonians, which includes those which we term "nearly-frustration-free": the class of Hamiltonians for which the ground state energy of their block-encoded and hence normalized Hamiltonian α−1H is within δy of -1, where δ is the spectral gap of α−1H and 0≤y≤1 . For this subclass, we describe an algorithm whose dependence on the gap is asymptotically better, scaling as δy/2−1 , and show that this new dependence is optimal up to factors of log⁡δ . In addition, we give examples of physically motivated Hamiltonians which live in this subclass. Finally, we describe an extension of this method which allows the preparation of excited states both for generic Hamiltonians as well as, at a similar speedup as the ground state case, for those which are nearly frustration-free.
The exponential speedups promised by Hamiltonian simulation on a quantum computer depends crucially on structure in both the Hamiltonian H^\hat{H}H^, and the quantum circuit U^\hat{U}U^ that encodes its description. In the quest to better approximate time-evolution e−iH^te^{-i\hat{H}t}e−iH^t with error εεε, we motivate a systematic approach to understanding and exploiting structure, in a setting where Hamiltonians are encoded as measurement operators of unitary circuits U^\hat{U}U^ for generalized measurement. This allows us to define a \emph{uniform spectral amplification} problem on this framework for expanding the spectrum of encoded Hamiltonian with exponentially small distortion. We present general solutions to uniform spectral amplification in a hierarchy where factoring U^\hat{U}U^ into n=1,2,3n=1,2,3n=1,2,3 unitary oracles represents increasing structural knowledge of the encoding. Combined with structural knowledge of the Hamiltonian, specializing these results allow us simulate time-evolution by ddd-sparse Hamiltonians using O(t(d∥H^∥max∥H^∥1)1/2log⁡(t∥H^∥/ε))\mathcal{O}\left(t(d \|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2}\log{(t\|\hat{H}\|/ε)}\right)O(t(d∥H^∥max​∥H^∥1​)1/2log(t∥H^∥/ε)) queries, where ∥H^∥≤∥H^∥1≤d∥H^∥max\|\hat H\|\le \|\hat H\|_1\le d\|\hat H\|_{\text{max}}∥H^∥≤∥H^∥1​≤d∥H^∥max​. Up to logarithmic factors, this is a polynomial improvement upon prior art using O(td∥H^∥max+log⁡(1/ε)log⁡log⁡(1/ε))\mathcal{O}\left(td\|\hat H\|_{\text{max}}+\frac{\log{(1/ε)}}{\log\log{(1/ε)}}\right)O(td∥H^∥max​+loglog(1/ε)log(1/ε)​) or O(t3/2(d∥H^∥max∥H^∥1∥H^∥/ε)1/2)\mathcal{O}(t^{3/2}(d \|\hat H\|_{\text{max}}\|\hat H\|_{1}\|\hat H\|/ε)^{1/2})O(t3/2(d∥H^∥max​∥H^∥1​∥H^∥/ε)1/2) queries. In the process, we also prove a matching lower bound of Ω(t(d∥H^∥max∥H^∥1)1/2)Ω(t(d\|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2})Ω(t(d∥H^∥max​∥H^∥1​)1/2) queries, present a distortion-free generalization of spectral gap amplification, and an amplitude amplification algorithm that performs multiplication on unknown state amplitudes.
The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the LLL computation steps and influencing the scaling of the necessary evolution time by its spectral gap of O(1L2)\mathcal{O}\left(\frac{1}{L^2}\right)O(L21​). Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap O(1Lk)\mathcal{O}\left(\frac{1}{L^k}\right)O(Lk1​), k<2k<2k<2, turn out to contradict very likely, graphs with k<1k<1k<1 even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.
Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps LLL. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.